Bi-level control for batch arrival queues with an early startup and un-reliable server

摘要

This paper studies a like-queue production system under bi-level control policy, where an unreliable server (machine) operates N policy with an early startup. The server may become idle or take vacations when system is empty. We model this system by M~([x])/G/1 queue under the following two situations: (1) As soon as there are no units (customers) to process, the server is turned off and becomes idle until m or more units are accumulated in the queue, at which time the server performs a startup with random length. After the startup, if there are N or more units waiting for processing, the server immediately begins serving the accumulated units. Otherwise the server remains dormant in the system and waits until the accumulated number of units reaches or exceeds N. (2) As soon as there are no units to process, the server is turned off and takes a vacation with random length. When he returns from the vacation and finds the number of waiting units is less than a predetermined threshold m, he takes a vacation again. Thus the server takes vacations repeatedly until he finds m or more waiting units in the queue. At that time the server is immediately turned on and requires a startup time. After the startup, if the system size is less than N, the server remains dormant in the system and waits until the accumulated number of units reaches or exceeds N. Further, it is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution. For both cases, we obtain the probability generating function in the system through the decomposition property and then derive the system characteristics.